Subgroups — #11756449

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#91786	C₂	1	2	yes	maximal, complemented
3	#91785	C₂	1	2	yes	maximal, complemented
4	#91785	C₂	1	2	yes	maximal, complemented
5	#91782	C₂	1	1	yes	maximal, characteristic
6	#91782	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×D₁₅	C₂	1	2	yes	maximal, complemented
8	#91779	C₂	1	1	yes	maximal, characteristic, complemented
9	#91777	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#66024	—	3	1	no	maximal, Hall
13	C₂×D₆₀	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅	C₂²	1	2	yes	—
15	C₂×C₄×D₁₅	C₂²	1	2	yes	—
16	C₂×C₄×D₁₅	C₂²	1	2	yes	—
17	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₀	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	#6714	C₂²	1	2	yes	—
25	#6714	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₁₅	C₂²	1	2	yes	—
30	#6712	C₂²	1	1	yes	characteristic
31	#6712	C₂²	1	1	yes	characteristic
32	#6712	C₂²	1	2	yes	—
33	#6712	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5690	—	5	1	no	maximal, Hall
37	C₅×#125	D₃	1	1	yes	characteristic, complemented
38	#4043	—	3	2	no	—
39	#4042	—	3	2	no	—
40	#4042	—	3	2	no	—
41	#4039	—	3	1	no	—
42	#4039	—	3	1	no	—
43	D₅×C₄⋊C₄	—	3	2	no	—
44	#4036	—	3	1	no	—
45	#4034	—	3	1	no	—
46	C₄×D₂₀	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂³	1	2	yes	—
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	D₆₀	Q₈	1	4	yes	complemented
60	C₄×D₁₅	—	2	4	no	—
61	C₄×D₁₅	—	4	2	no	—
62	C₄×D₁₅	—	4	2	no	—
63	Dic₃₀	—	2	2	no	—
64	Dic₃₀	—	4	2	no	—
65	C₃×#125	D₅	1	1	yes	characteristic, complemented
66	#897	—	5	2	no	—
67	#896	—	5	2	no	—
68	#896	—	5	2	no	—
69	#893	—	5	1	no	—
70	#893	—	5	1	no	—
71	D₃×C₄⋊C₄	—	5	2	no	—
72	#890	—	5	1	no	—
73	#888	—	5	1	no	—
74	C₄×D₁₂	—	5	1	no	—
75	C₄×Dic₆	—	5	1	no	—
76	C₂×D₂₀	—	3	1	no	—
77	C₂×C₄×D₅	—	3	2	no	—
78	C₂×C₄×D₅	—	3	2	no	—
79	C₂×C₄×D₅	—	3	2	no	—
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
84	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
85	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
86	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
87	#685	—	3	2	no	—
88	#685	—	3	2	no	—
89	#685	—	3	2	no	—
90	C₄⋊Dic₅	—	3	1	no	—
91	C₄⋊Dic₅	—	3	1	no	—
92	C₄⋊Dic₅	—	3	2	no	—
93	#683	—	3	1	no	—
94	#683	—	3	1	no	—
95	#683	—	3	2	no	—
96	#683	—	3	2	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₄×Dic₅	—	3	1	no	—
99	#554	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×D₁₅	—	2	2	no	—
102	C₂×D₁₅	—	2	2	no	—
103	C₂×D₁₅	C₂×Q₈	1	4	yes	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
109	Dic₁₅	—	2	1	no	—
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	—	2	2	no	—
112	Dic₁₅	—	2	1	no	—
113	Dic₁₅	#41	1	2	yes	—
114	C₂×D₁₂	—	5	1	no	—
115	C₂×C₄×D₃	—	5	2	no	—
116	C₂×C₄×D₃	—	5	2	no	—
117	C₂×C₄×D₃	—	5	2	no	—
118	C₂×Dic₆	—	5	1	no	—
119	C₂×Dic₆	—	5	2	no	—
120	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
121	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
123	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
124	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
125	(C₂×D₃)⋊C₄	—	5	2	no	—
126	(C₂×D₃)⋊C₄	—	5	2	no	—
127	(C₂×D₃)⋊C₄	—	5	2	no	—
128	C₄⋊Dic₃	—	5	1	no	—
129	C₄⋊Dic₃	—	5	1	no	—
130	C₄⋊Dic₃	—	5	2	no	—
131	Dic₃⋊C₄	—	5	1	no	—
132	Dic₃⋊C₄	—	5	1	no	—
133	Dic₃⋊C₄	—	5	2	no	—
134	Dic₃⋊C₄	—	5	2	no	—
135	C₄×Dic₃	—	5	1	no	—
136	C₄×Dic₃	—	5	1	no	—
137	C₂²×D₅	—	3	2	no	—
138	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	1	no	—
147	C₂×Dic₅	—	3	2	no	—
148	D₂₀	—	3	4	no	—
149	C₄×D₅	—	6	4	no	—
150	C₄×D₅	—	12	2	no	—
151	C₄×D₅	—	12	2	no	—
152	Dic₁₀	—	6	2	no	—
153	Dic₁₀	—	12	2	no	—
154	C₄⋊Q₈	—	15	1	no	—
155	#125	—	15	1	no	—
156	#125	D₁₅	1	1	yes	characteristic, complemented
157	C₂²⋊Q₈	—	15	2	no	—
158	C₂²⋊Q₈	—	15	2	no	—
159	C₂²⋊Q₈	—	15	2	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	15	1	no	—
162	C₄×D₄	—	15	1	no	—
163	C₄×D₄	—	15	1	no	—
164	C₂×C₄⋊C₄	—	15	2	no	—
165	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	D₁₅	—	2	4	no	—
169	C₂²×D₃	—	5	2	no	—
170	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×Dic₃	—	5	1	no	—
176	C₂×Dic₃	—	5	1	no	—
177	C₂×Dic₃	—	5	1	no	—
178	C₂×Dic₃	—	5	1	no	—
179	C₂×Dic₃	—	5	2	no	—
180	D₁₂	—	5	4	no	—
181	C₄×D₃	—	10	4	no	—
182	C₄×D₃	—	20	2	no	—
183	C₄×D₃	—	20	2	no	—
184	Dic₆	—	10	2	no	—
185	Dic₆	—	20	2	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₂×D₅	—	3	4	no	—
188	C₂×D₅	—	6	2	no	—
189	C₂×D₅	—	6	2	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	D₃×Q₈	1	2	yes	—
195	Dic₅	—	3	2	no	—
196	Dic₅	—	6	1	no	—
197	Dic₅	—	6	1	no	—
198	Dic₅	—	6	2	no	—
199	Dic₅	—	6	1	no	—
200	C₂×Q₈	—	15	1	no	—
201	C₂×Q₈	—	15	2	no	—
202	C₂×D₄	—	15	1	no	—
203	C₂²×C₄	—	15	2	no	—
204	C₂²×C₄	—	15	2	no	—
205	C₂²×C₄	—	15	2	no	—
206	C₄⋊C₄	—	15	1	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊C₄	—	15	1	no	—
211	C₄⋊C₄	—	15	1	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
214	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
215	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
216	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
217	C₂²⋊C₄	—	15	2	no	—
218	C₂²⋊C₄	—	15	2	no	—
219	C₂²⋊C₄	—	15	2	no	—
220	C₄²	—	15	1	no	—
221	C₄²	—	15	1	no	—
222	C₄²	C₂×D₁₅	1	1	yes	characteristic
223	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
225	C₂×D₃	—	5	4	no	—
226	C₂×D₃	—	10	2	no	—
227	C₂×D₃	—	10	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	Q₈×D₅	1	2	yes	—
233	Dic₃	—	5	2	no	—
234	Dic₃	—	10	1	no	—
235	Dic₃	—	10	1	no	—
236	Dic₃	—	10	2	no	—
237	Dic₃	—	10	1	no	—
238	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
239	C₂×C₅	D₃×#41	1	1	yes	characteristic
240	C₂×C₅	#1009	1	1	yes	characteristic
241	D₅	—	6	4	no	—
242	C₂³	—	15	2	no	—
243	Q₈	—	30	2	no	—
244	Q₈	—	60	2	no	—
245	D₄	—	15	4	no	—
246	C₂×C₄	—	15	1	no	—
247	C₂×C₄	—	15	1	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	15	1	no	—
250	C₂×C₄	—	15	2	no	—
251	C₂×C₄	—	30	4	no	—
252	C₂×C₄	—	60	2	no	—
253	C₂×C₄	—	60	2	no	—
254	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₁₅	1	2	yes	—
257	C₂×C₄	C₂²×D₁₅	1	2	yes	—
258	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
259	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
260	C₂×C₃	D₅×#41	1	1	yes	characteristic
261	C₂×C₃	#4149	1	1	yes	characteristic
262	D₃	—	10	4	no	—
263	C₅	#5690	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	15	4	no	—
265	C₂²	—	30	2	no	—
266	C₂²	—	30	2	no	—
267	C₂²	C₂³×D₁₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	15	2	no	—
273	C₄	—	30	1	no	—
274	C₄	—	30	1	no	—
275	C₄	—	30	2	no	—
276	C₄	—	30	1	no	—
277	C₄	Q₈×D₁₅	1	2	yes	—
278	C₃	#66024	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	30	4	no	—
280	C₂	C₂×Q₈×D₁₅	1	1	yes	characteristic
281	C₂	#41×D₁₅	1	1	yes	characteristic
282	C₂	#92100	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×#125	C₂
prime radical	C₂×D₆₀	C₂²
hypercenter	#125	D₁₅
hypocenter	C₃×C₅	#554
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₅×#125	C₂	semidirect
2. #91786	C₂	semidirect
3. #91785	C₂	semidirect
4. #91785	C₂	semidirect
7. C₄⋊C₄×D₁₅	C₂	semidirect
8. #91779	C₂	semidirect
9. #91777	C₂	semidirect
11. C₄×Dic₃₀	C₂	semidirect
37. C₅×#125	D₃	semidirect
59. D₆₀	Q₈	semidirect
65. C₃×#125	D₅	semidirect
156. #125	D₁₅	semidirect
223. C₃×C₅	#554	semidirect
263. C₅	#5690	semidirect
278. C₃	#66024	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#125	2
#91786	2
#91785	2
#91782	2
C₄⋊C₄×D₁₅	2
#91779	2
#91777	2
C₄×D₆₀	2
C₄×Dic₃₀	2
#66024	3
#5690	5